To verify that the equation

$\csc(x) - \sin(x) = \cos(x) \cdot \cot(x)$

is an identity, let's work through both sides and check if they are equivalent.

Step 1: Rewrite Both Sides Using Trigonometric Identities

1. Left Side: $\csc(x) - \sin(x)$

   Recall that:

   • $\csc(x) = \frac{1}{\sin(x)}$

   Substituting, we get: $\csc(x) - \sin(x) = \frac{1}{\sin(x)} - \sin(x)$

   We can combine terms by expressing $\sin(x)$ as $\frac{\sin^2(x)}{\sin(x)}$: $\frac{1}{\sin(x)} - \sin(x) = \frac{1 - \sin^2(x)}{\sin(x)}$

   Next, use the Pythagorean identity $1 - \sin^2(x) = \cos^2(x)$: $\frac{1 - \sin^2(x)}{\sin(x)} = \frac{\cos^2(x)}{\sin(x)}$

   So, the left side simplifies to: $\frac{\cos^2(x)}{\sin(x)}$

2. Right Side: $\cos(x) \cdot \cot(x)$

   Recall that:

   • $\cot(x) = \frac{\cos(x)}{\sin(x)}$

   Substituting, we get: $\cos(x) \cdot \cot(x) = \cos(x) \cdot \frac{\cos(x)}{\sin(x)} = \frac{\cos^2(x)}{\sin(x)}$

Conclusion

Both the left side and right side simplify to $\frac{\cos^2(x)}{\sin(x)}$, so:

$\csc(x) - \sin(x) = \cos(x) \cdot \cot(x)$

Thus, the equation is indeed an identity.

Would you like further details on any part of this solution?

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Here are five related questions to consider:

1. What are some other common trigonometric identities?
2. How do you simplify expressions involving $\csc(x)$ or $\cot(x)$?
3. Can this identity be derived using other trigonometric identities?
4. Why is the Pythagorean identity useful in simplifying trigonometric expressions?
5. How would you verify a trigonometric identity if both sides have multiple trigonometric functions?

Tip: When verifying trigonometric identities, try to express everything in terms of $\sin(x)$ and $\cos(x)$ — this often simplifies the process!